Why is it appropriate to use a firm's weighted average cost of capital to evaluate investment projects?

Because that's a great indicator of the firm's reliability. This generally only applies in ultra volatile markets though.

Let's make this simple, it is the same reason you use average costs for any type of valuation. It is to minimize the monetary costs of mismeasuring the value of the project. If you use the highest costs, you might overspend or over-allocate or make a measurement error that exceeds the actual cost (expected>actual) if you use the highest costs. If you use the lowest costs, you may make a measurement error that is less than the actual cost (expected<actual). That is why you just keep the average, it mitigates the risk of measurement error.

The cost of capital is the return a company has to get on an investment in order to maintain the value of the company.

It is appropriate because that is the rate that investors' require the company to earn in order to reimburse them for the risk inherent in investing in that company. The weighted average cost of capital simply combines the cost of equity and the cost of debt (and other sources of capital) based upon their relative portion of total invested capital. This creates a blended rate that is used to measure the required return to all capital holders in that company.
If a company can not invest in programs that return at or above the WACC, then investment theory dictates the money should be returned to shareholders so they can find other investment opportunities for their capital that will appropriately reimburse them for the risk they are accepting. In reality, corporate investments are made with inadequate information that often decreases the real rate of return from invested projects. Researchers believe that there are many reasons for this including: agency problems, inaccurate forecasts, and decision-making biases amongst others.